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Results tagged with real-analysis
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user 47322
Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
2
votes
Monotonicity of the sequences of the lower and upper Darboux sums
This is a long comment to @AnthonyQuas's solution. It works out the details of the stochastic dominance as stated by Anthony.
Let $F_n(x) = \int_{-\infty}^x \,d\mu^+_n(t)$ be the cumulative distrib …
- 489
0
votes
Is the domain of symmetric derivative borel set?
We can majorize $\lambda $ by a non-negative Borel measure $|\lambda|$. Let $\nu = |\lambda| + \mu .$ Since $|\lambda|$ and $\mu$ are absolutly continuos with respect to $\nu$, we can represent $\lam …
- 489
2
votes
Checking $f(x_1,y_1)f(x_2,y_2)-f(x_1,y_2)f(x_2,y_1) \ge 0$
To follow up @igor answer, the equation can be written as:
$$ \dfrac{\partial^2 \ln f}{\partial x \partial y} \geq 0.
$$
Moreover, the inequality is not only necessary, bu sufficient, since:
$$
\ln \ …
- 489
1
vote
Accepted
Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$
Let's pursue Jochen's idea. We assume $A \ne \emptyset.$
Let
$$ \varphi(t) = \begin{cases}
e^{-\frac{1}{t}} &\text{if $ t>0$}\\
0 &\text{otherwise.}
\end{cases}$$
This func …
- 489